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Many problems in multivariate analysis involve the $n \times n$ matrix:

$$\mathbf{M} \equiv \boldsymbol{I}_n - \frac{1}{n} \mathbf{1}_{n \times n}.$$

This is an orthogonal projection matrix onto the unit vector, so when it is applied to a column vector containing values $Y_1,...,Y_n$, this matrix subtracts the sample mean $\bar{Y}_n$ from these values:

$$\mathbf{Y} = \begin{bmatrix} Y_1 \\ \vdots \\ Y_n \end{bmatrix} \quad \quad \quad \implies \quad \quad \quad \mathbf{M} \mathbf{Y} = \begin{bmatrix} Y_1 - \bar{Y}_n \\ \vdots \\ Y_n - \bar{Y}_n \end{bmatrix}.$$

There are many simple properties of this matrix, owing to the fact that it is an orthogonal projection matrix. It has $\text{tr}(\mathbf{M}) = \text{rank}(\mathbf{M}) = n-1$, which means that it has a single zero eigenvalue and the remaining eigenvalues are all ones. It comes up a lot in multivariate analysis, including statistical problems, where it is common to look at random vectors after subtracting their sample means.

My question: Does this matrix have any special name? Is there a literature on this type of matrix? Aside from the properties I have listed here, are there any other important properties of this matrix that are useful in statistical problems dealing with random vectors?

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  • $\begingroup$ Maybe look at MDS or PCA literature. I vaguely recall it being called the mean-centering matrix there. $\endgroup$ – obscurans Jan 31 '19 at 2:59
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From the comment by obscurans I have now found out that this is called the centering matrix. It appears to be discussed specifically in some matrix textbooks, such as in Zhang (2017) (pp. 111-112). It does not appear to have any other important properties beyond what was listed in the question.

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